SAGA winter school Rational cuspidal plane curves Torgunn Karoline ...
SAGA winter school Auron, France March 2010
Rational cuspidal plane curves
Torgunn Karoline Moe Centre of Mathematics for Applications University of Oslo, Norway
Torgunn Karoline Moe
Cuspidal curves Introduction
Rational cuspidal plane curves
Background Cuspidal curves Cremona Projections Conjecture Real cusps References
• How many and what kind of cusps can a rational
cuspidal plane curve have?
Torgunn Karoline Moe
Cuspidal curves Introduction
Overview Background Cuspidal curves
• Background
Projections
• Rational cuspidal plane curves of degree 3 and 4 • Constructions • Cremona transformations • Projections
Conjecture
• Rational cuspidal plane curves with many cusps
Real cusps
• The real perspective
Cremona
References
Torgunn Karoline Moe
Cuspidal curves Introduction Background Cuspidal curves Cremona Projections Conjecture Real cusps References
Background • The projective plane P2 over the complex numbers C
with coordinates (x : y : z). • A curve in P2 given by V(F ), where F ∈ C[x, y , z]. • A point a = (a0 : a1 : a2 ) on a curve V(F ) is called
singular if it is in the zero set of all the partial derivatives of F , • a ∈ V(Fx , Fy , Fz ).
• A curve can only have a finite number of singular
points. • Every singular point p has one or more tangent line(s). • For p = (0 : 0 : 1) singular,
F (x, y , 1) = fm (x, y ) + fm+1 (x, y ) + . . . + fd (x, y ). • The tangent line(s) of C at p is given by the zero
set(s) of each reduced linear factor of fm (x, y ).
Torgunn Karoline Moe
Cuspidal curves
Examples of singularities on curves
Introduction Background Cuspidal curves Cremona Projections Conjecture Real cusps References
The nodal cubic V(zx 2 − zy 2 − x 3 )
The cuspidal cubic V(zy 2 − x 3 )
• How and why are these different?
Torgunn Karoline Moe
Cuspidal curves Introduction Background Cuspidal curves
Invariants of singular points • Branches – the number of times the curve passes through the point. • A singularity with more than one branch is called a
multiple point. • A singularity with only one branch is called a cusp.
Cremona Projections Conjecture Real cusps References
• Multiplicity – the intersection multiplicity of a
general line and the curve at the point. • Is equal to the m in fm (x, y ) for p = (0 : 0 : 1).
• Tangent intersection – the intersection multiplicity of
the tangent line and the curve at the point. • Can investigate the inside of a singularity by blowing
it up, if necessary, several times.
Torgunn Karoline Moe
Cuspidal curves Introduction Background Cuspidal curves Cremona Projections Conjecture Real cusps References
Blowing up the cuspidal cubic
Torgunn Karoline Moe
Cuspidal curves Introduction Background Cuspidal curves Cremona Projections Conjecture Real cusps References
Torgunn Karoline Moe
Cuspidal curves Introduction
The multiplicity sequence of a cusp
Background
• Let mi denote the multiplicity of the remaining
Cuspidal curves
singularity after i blowing-ups. • For a cusp we have the multiplicity sequence m
Cremona Projections
• m = (m, m1 , . . . , ms ). • m≥m1 ≥ . . . ≥ms . • For every i there is a k ≥ 0 such that
Conjecture
mi−1
Real cusps
mi−1
References
= mi + . . . + mi+k , where mi =mi+1 = . . . =mi+k−1 . = k · mi + mi+k .
Torgunn Karoline Moe
Rational cuspidal plane curves Cuspidal curves Introduction Background Cuspidal curves Cremona Projections Conjecture
• A curve is called cuspidal if all its singularities are
cusps. • A (cuspidal) curve of degree d is rational ⇐⇒ the
genus formula holds. (d − 1)(d − 2) = 2
X singular points
X mi (mi − 1) ( ). 2 i
• How many and what kind of cusps can a rational
cuspidal curve have? • By the genus formula, the cuspidal cubic is the only
Real cusps References
rational cuspidal curve of degree 3. • A rational cuspidal plane curve of degree d must also
satisfy • B´ ezout: mp + mq ≤ d. • Matsuoka–Sakai: d < 3 · m,
where m is the highest multiplicity of the cusps.
Torgunn Karoline Moe
Rational cuspidal curves of degree 4
Cuspidal curves Introduction Background Cuspidal curves Cremona (2), (2), (2)
(22 ), (2)
Projections Conjecture Real cusps References
(3)
(23 )
Torgunn Karoline Moe
Cuspidal curves
Cremona transformations • Birational maps ψ,
S
Introduction Background Cuspidal curves Cremona Projections Conjecture Real cusps References
. ψ : P2 ∪ C
& 99K 7−→
P2 ∪ C0
• ψ : (x : y : z) 7−→ (G0 : G1 : G2 ), • G0 , G1 , G2 homogeneous polynomials in x, y , z of
the same degree n. • Linearly independent and without common factors. • V(G0 , G1 , G2 ) consists of n2 − 1 points.
• A linear transformation is a Cremona transformation. • A curve C = V(F ) is mapped to a total transform,
whose defining polynomial consists of • linear factors, • the defining polynomial of the strict transform C 0 .
Torgunn Karoline Moe
Cuspidal curves Introduction Background Cuspidal curves Cremona Projections Conjecture Real cusps References
• A general Cremona transformation involves blowing
up points and contracting lines. • Resolves a singularity on C by blowing it up. • Creates a cusp on C 0 by contracting a tangent line
of C that only intersects it in one point p. • The multiplicity of the cusp p 0 on C 0 equals the
intersection multiplicity of the tangent and C in p. • Note that we can create other singularities by
contracting other lines.
Torgunn Karoline Moe
Cuspidal curves
A quadratic Cremona transformation Introduction Background Cuspidal curves
• Blows up three points p, q, r . • Contracts three lines. • Note that the points and the lines do not have to be in P2 .
Cremona
• The strict transform C 0 has degree d 0 ,
Projections
d 0 = 2 · d − mp − mq − mr . • There are essentially three different quadratic Cremona transformations:
Conjecture Real cusps References
• ψ3 : (x : y : z) 7−→ (yz : xz : xy ). • ψ2 : (x : y : z) − 7 → (xy : z 2 : yz). • ψ1 : (x : y : z) 7−→ (y 2 − xz : yz : z 2 ).
Torgunn Karoline Moe
Cuspidal curves Introduction Background Cuspidal curves Cremona Projections Conjecture Real cusps References
The cuspidal cubic – [(2)]
Torgunn Karoline Moe
Cuspidal curves Introduction Background Cuspidal curves Cremona Projections Conjecture Real cusps References
The cuspidal cubic – [(2)]
Torgunn Karoline Moe
Cuspidal curves Introduction Background Cuspidal curves Cremona Projections Conjecture Real cusps References
The cuspidal cubic – [(2)]
Torgunn Karoline Moe
Cuspidal curves
The cuspidal cubic – [(2)]
Introduction Background Cuspidal curves Cremona Projections Conjecture Real cusps References
V(x 2 − 2xy + y 2 + xz + yz) 7−→ V(y 2 z − 2xyz + x 2 z + xy 2 + x 2 y )
Torgunn Karoline Moe
Cuspidal curves Introduction Background Cuspidal curves Cremona Projections Conjecture Real cusps References
The tricuspidal quartic – [(2), (2), (2)]
Torgunn Karoline Moe
Cuspidal curves Introduction Background Cuspidal curves Cremona Projections Conjecture Real cusps References
The tricuspidal quartic – [(2), (2), (2)]
Torgunn Karoline Moe
Cuspidal curves Introduction Background Cuspidal curves Cremona Projections Conjecture Real cusps References
The tricuspidal quartic – [(2), (2), (2)]
Torgunn Karoline Moe
Cuspidal curves
The tricuspidal quartic – [(2), (2), (2)]
Introduction Background Cuspidal curves Cremona Projections Conjecture Real cusps References
V(x 2 + y 2 + z 2 − 2xy − 2xz − 2yz) 7−→ V(y 2 z 2 + x 2 z 2 + x 2 y 2 − 2xyz 2 − 2xy 2 z − 2x 2 yz)
Torgunn Karoline Moe
Cuspidal curves Introduction Background Cuspidal curves Cremona
The power of Cremona transformations • It is possible to construct the other curves of degree
4 in similar ways. • Choosing the points and lines of the transformation
Projections
and the orientation of the curve C is sometimes complicated.
Conjecture
• [Fenske, Flenner & Zaidenberg (1996-1999)] used
Real cusps References
Cremona transformations to construct infinite series of cuspidal curves from cuspidal curves of low degree.
Torgunn Karoline Moe
Cuspidal curves Introduction Background Cuspidal curves Cremona Projections Conjecture Real cusps References
Projections φV : Pn 99K P2 ∪ ∪ C − 7 → C0 • C curve in Pn . • V a linear subspace in Pn of dimension (n − 3), called the centre of projection. • C 0 the image of C in P2 .
Torgunn Karoline Moe
Cuspidal curves
A special curve and some of its properties • The curve Cn in Pn given by
Introduction Background Cuspidal curves Cremona Projections
(s n : s n−1 t : . . . : st n−1 : t n ). is called the rational normal curve of degree n. • This is a smooth curve, and all points have the same
properties. • For each pair (s : t) corresponding to a point on Cn ,
Conjecture
the curve has a unique tangent line given by the linear span of
Real cusps
References
ns n−1 (n − 1)s n−2 t . . . t n−1 0 . 0 s n−1 . . . (n − 1)st n−2 nt n−1
• The union of all the tangents is a surface called the
tangent developable T of Cn .
Torgunn Karoline Moe
Cuspidal curves Introduction Background
• Two points on Cn define a unique line in Pn .
Cuspidal curves
• Such a line is called a secant line.
Cremona
• The union of all the secant lines is a threefold called
Projections Conjecture Real cusps References
the secant variety S of Cn . • The tangent developable is contained in the secant
variety.
Torgunn Karoline Moe
Cuspidal curves Introduction Background Cuspidal curves Cremona Projections Conjecture Real cusps References
Cuspidal projections • A rational cuspidal plane curve C 0 of degree n can be constructed from Cn using a suitable centre of projection V : • V can not intersect the curve Cn . • That would reduce the degree of C 0 . • V must intersect the tangent developable T . • Get one cusp on C 0 for every intersection point of V and T . • V can not intersect the secant variety S outside T . • That would give other singularities.
• We use Cn and the kernel of V written on matrix
form to find the parametrization of C 0 . • The defining polynomial of C 0 can be found by
eliminating s and t.
Torgunn Karoline Moe
Cuspidal curves Introduction Background Cuspidal curves Cremona Projections Conjecture Real cusps References
The cuspidal cubic • C3 = (s 3 : s 2 t : st 2 : t 3 ). • The centre of projection V is a point on the tangent
developable. • Can choose V = (0 : 1 : 0 : 0). • Parametrization – C 0 = (s 3 : st 2 : t 3 ). • Defining equation – C 0 = V(y 3 − xz 2 ).
Torgunn Karoline Moe
Cuspidal curves Introduction Background
The tricuspidal quartic • C4 = (s 4 : s 3 t : s 2 t 2 : st 3 : t 4 ). • The centre of projection V is a line which intersects
Cuspidal curves Cremona Projections Conjecture Real cusps References
the tangent developable in three points. • V can be found using the appropriate conditions.
V =
2 1 0 0 0 . 0 0 0 1 2
• C 0 = (s 3 t − 12 s 4 : s 2 t 2 : t 4 − 2st 3 ). • C 0 = V(8xy 3 − 3y 4 + 12xy 2 z − 4y 3 z + 4x 2 z 2 ).
Torgunn Karoline Moe
Cuspidal curves Introduction
Rational cuspidal curves of degree 5
Background
# Cusps
Cuspidal curves
1
Cremona Projections
2
Conjecture
3
Real cusps
4
References
Curve C1 C2 C3 C4 C5 C6 C7 C8
Cuspidal configuration (4) (26 ) (3, 2), (22 ) (3), (23 ) (24 ), (22 ) (3), (22 ), (2) (22 ), (22 ), (22 ) (23 ), (2), (2), (2)
# Curves 3 – ABC 1 2 – AB 1 1 1 1 1
Torgunn Karoline Moe
Cuspidal curves
Conjecture
Introduction
• There is only one rational cuspidal plane curve with
Background
more than three cusps – the curve of degree 5 with cuspidal configuration [(23 ), (2), (2), (2)]. • [Piontkowski (2007)] The only tricuspidal curves are
Cuspidal curves
• [Fenske, Flenner & Zaidenberg (1996-1999)]
Cremona
Series I II III
Projections Conjecture
m ¯p (d − 2) (d − 3, 2a ) (d − 4, 3a )
m ¯q (2a ) (3a ) (4a , 22 )
m ¯r (2d−2−a ) (2) (2)
For d d ≥4 d ≥5 d ≥7
• The curve of degree 5 with cuspidal configuration
[(22 ), (22 ), (22 )].
Real cusps References
d d 2a + 3 3a + 4
Result • [Tono (2005)] A rational cuspidal curve has ≤ 8
cusps.
Torgunn Karoline Moe
Cuspidal curves Introduction Background Cuspidal curves Cremona
Can all cusps be given real coordinates? • By the conjecture, a rational cuspidal curve will never have more than four cusps. • A linear transformation can move three cusps to real
coordinates. Projections Conjecture Real cusps References
• There is only one curve left to consider,
[(23 ), (2), (2), (2)].
Torgunn Karoline Moe
Cuspidal curves
• For real cuspidal curves, we have a modified Introduction Background Cuspidal curves
Klein–Schuh equality involving the curve C , its cusps and its dual curve C ∗ . d−
X real cusps on C
Cremona Projections Conjecture Real cusps References
(m − 1) = d ∗ −
X real cusps on
(m∗ − 1) C∗
• The dual of [(23 ), (2), (2), (2)] is the quartic [(23 )]. • Assuming the quintic is a real curve, and assuming
its cusps have real coordinates leads to a contradiction of Klein–Schuh. • Can the defining polynomial of [(23 ), (2), (2), (2)]
have complex coefficients while the cusps have real coordinates?
Torgunn Karoline Moe
Cuspidal curves Introduction
References T. Fenske Rational cuspidal plane curves of type (d, d − 4) with χ(ΘV hDi) ≤ 0.
Background Cuspidal curves Cremona Projections Conjecture
H. Flenner, M. Zaidenberg On a class of rational cuspidal plane curves. H. Flenner, M. Zaidenberg Rational cuspidal plane curves of type (d, d − 3). M. Namba. Geometry of projective algebraic curves.
Real cusps References
J. Piontkowski. On the Number of the Cusps of Rational Cuspidal Plane Curves. K. Tono On the number of the cusps of cuspidal plane curves.
Rational cuspidal plane curves
Torgunn Karoline Moe Centre of Mathematics for Applications University of Oslo, Norway
Torgunn Karoline Moe
Cuspidal curves Introduction
Rational cuspidal plane curves
Background Cuspidal curves Cremona Projections Conjecture Real cusps References
• How many and what kind of cusps can a rational
cuspidal plane curve have?
Torgunn Karoline Moe
Cuspidal curves Introduction
Overview Background Cuspidal curves
• Background
Projections
• Rational cuspidal plane curves of degree 3 and 4 • Constructions • Cremona transformations • Projections
Conjecture
• Rational cuspidal plane curves with many cusps
Real cusps
• The real perspective
Cremona
References
Torgunn Karoline Moe
Cuspidal curves Introduction Background Cuspidal curves Cremona Projections Conjecture Real cusps References
Background • The projective plane P2 over the complex numbers C
with coordinates (x : y : z). • A curve in P2 given by V(F ), where F ∈ C[x, y , z]. • A point a = (a0 : a1 : a2 ) on a curve V(F ) is called
singular if it is in the zero set of all the partial derivatives of F , • a ∈ V(Fx , Fy , Fz ).
• A curve can only have a finite number of singular
points. • Every singular point p has one or more tangent line(s). • For p = (0 : 0 : 1) singular,
F (x, y , 1) = fm (x, y ) + fm+1 (x, y ) + . . . + fd (x, y ). • The tangent line(s) of C at p is given by the zero
set(s) of each reduced linear factor of fm (x, y ).
Torgunn Karoline Moe
Cuspidal curves
Examples of singularities on curves
Introduction Background Cuspidal curves Cremona Projections Conjecture Real cusps References
The nodal cubic V(zx 2 − zy 2 − x 3 )
The cuspidal cubic V(zy 2 − x 3 )
• How and why are these different?
Torgunn Karoline Moe
Cuspidal curves Introduction Background Cuspidal curves
Invariants of singular points • Branches – the number of times the curve passes through the point. • A singularity with more than one branch is called a
multiple point. • A singularity with only one branch is called a cusp.
Cremona Projections Conjecture Real cusps References
• Multiplicity – the intersection multiplicity of a
general line and the curve at the point. • Is equal to the m in fm (x, y ) for p = (0 : 0 : 1).
• Tangent intersection – the intersection multiplicity of
the tangent line and the curve at the point. • Can investigate the inside of a singularity by blowing
it up, if necessary, several times.
Torgunn Karoline Moe
Cuspidal curves Introduction Background Cuspidal curves Cremona Projections Conjecture Real cusps References
Blowing up the cuspidal cubic
Torgunn Karoline Moe
Cuspidal curves Introduction Background Cuspidal curves Cremona Projections Conjecture Real cusps References
Torgunn Karoline Moe
Cuspidal curves Introduction
The multiplicity sequence of a cusp
Background
• Let mi denote the multiplicity of the remaining
Cuspidal curves
singularity after i blowing-ups. • For a cusp we have the multiplicity sequence m
Cremona Projections
• m = (m, m1 , . . . , ms ). • m≥m1 ≥ . . . ≥ms . • For every i there is a k ≥ 0 such that
Conjecture
mi−1
Real cusps
mi−1
References
= mi + . . . + mi+k , where mi =mi+1 = . . . =mi+k−1 . = k · mi + mi+k .
Torgunn Karoline Moe
Rational cuspidal plane curves Cuspidal curves Introduction Background Cuspidal curves Cremona Projections Conjecture
• A curve is called cuspidal if all its singularities are
cusps. • A (cuspidal) curve of degree d is rational ⇐⇒ the
genus formula holds. (d − 1)(d − 2) = 2
X singular points
X mi (mi − 1) ( ). 2 i
• How many and what kind of cusps can a rational
cuspidal curve have? • By the genus formula, the cuspidal cubic is the only
Real cusps References
rational cuspidal curve of degree 3. • A rational cuspidal plane curve of degree d must also
satisfy • B´ ezout: mp + mq ≤ d. • Matsuoka–Sakai: d < 3 · m,
where m is the highest multiplicity of the cusps.
Torgunn Karoline Moe
Rational cuspidal curves of degree 4
Cuspidal curves Introduction Background Cuspidal curves Cremona (2), (2), (2)
(22 ), (2)
Projections Conjecture Real cusps References
(3)
(23 )
Torgunn Karoline Moe
Cuspidal curves
Cremona transformations • Birational maps ψ,
S
Introduction Background Cuspidal curves Cremona Projections Conjecture Real cusps References
. ψ : P2 ∪ C
& 99K 7−→
P2 ∪ C0
• ψ : (x : y : z) 7−→ (G0 : G1 : G2 ), • G0 , G1 , G2 homogeneous polynomials in x, y , z of
the same degree n. • Linearly independent and without common factors. • V(G0 , G1 , G2 ) consists of n2 − 1 points.
• A linear transformation is a Cremona transformation. • A curve C = V(F ) is mapped to a total transform,
whose defining polynomial consists of • linear factors, • the defining polynomial of the strict transform C 0 .
Torgunn Karoline Moe
Cuspidal curves Introduction Background Cuspidal curves Cremona Projections Conjecture Real cusps References
• A general Cremona transformation involves blowing
up points and contracting lines. • Resolves a singularity on C by blowing it up. • Creates a cusp on C 0 by contracting a tangent line
of C that only intersects it in one point p. • The multiplicity of the cusp p 0 on C 0 equals the
intersection multiplicity of the tangent and C in p. • Note that we can create other singularities by
contracting other lines.
Torgunn Karoline Moe
Cuspidal curves
A quadratic Cremona transformation Introduction Background Cuspidal curves
• Blows up three points p, q, r . • Contracts three lines. • Note that the points and the lines do not have to be in P2 .
Cremona
• The strict transform C 0 has degree d 0 ,
Projections
d 0 = 2 · d − mp − mq − mr . • There are essentially three different quadratic Cremona transformations:
Conjecture Real cusps References
• ψ3 : (x : y : z) 7−→ (yz : xz : xy ). • ψ2 : (x : y : z) − 7 → (xy : z 2 : yz). • ψ1 : (x : y : z) 7−→ (y 2 − xz : yz : z 2 ).
Torgunn Karoline Moe
Cuspidal curves Introduction Background Cuspidal curves Cremona Projections Conjecture Real cusps References
The cuspidal cubic – [(2)]
Torgunn Karoline Moe
Cuspidal curves Introduction Background Cuspidal curves Cremona Projections Conjecture Real cusps References
The cuspidal cubic – [(2)]
Torgunn Karoline Moe
Cuspidal curves Introduction Background Cuspidal curves Cremona Projections Conjecture Real cusps References
The cuspidal cubic – [(2)]
Torgunn Karoline Moe
Cuspidal curves
The cuspidal cubic – [(2)]
Introduction Background Cuspidal curves Cremona Projections Conjecture Real cusps References
V(x 2 − 2xy + y 2 + xz + yz) 7−→ V(y 2 z − 2xyz + x 2 z + xy 2 + x 2 y )
Torgunn Karoline Moe
Cuspidal curves Introduction Background Cuspidal curves Cremona Projections Conjecture Real cusps References
The tricuspidal quartic – [(2), (2), (2)]
Torgunn Karoline Moe
Cuspidal curves Introduction Background Cuspidal curves Cremona Projections Conjecture Real cusps References
The tricuspidal quartic – [(2), (2), (2)]
Torgunn Karoline Moe
Cuspidal curves Introduction Background Cuspidal curves Cremona Projections Conjecture Real cusps References
The tricuspidal quartic – [(2), (2), (2)]
Torgunn Karoline Moe
Cuspidal curves
The tricuspidal quartic – [(2), (2), (2)]
Introduction Background Cuspidal curves Cremona Projections Conjecture Real cusps References
V(x 2 + y 2 + z 2 − 2xy − 2xz − 2yz) 7−→ V(y 2 z 2 + x 2 z 2 + x 2 y 2 − 2xyz 2 − 2xy 2 z − 2x 2 yz)
Torgunn Karoline Moe
Cuspidal curves Introduction Background Cuspidal curves Cremona
The power of Cremona transformations • It is possible to construct the other curves of degree
4 in similar ways. • Choosing the points and lines of the transformation
Projections
and the orientation of the curve C is sometimes complicated.
Conjecture
• [Fenske, Flenner & Zaidenberg (1996-1999)] used
Real cusps References
Cremona transformations to construct infinite series of cuspidal curves from cuspidal curves of low degree.
Torgunn Karoline Moe
Cuspidal curves Introduction Background Cuspidal curves Cremona Projections Conjecture Real cusps References
Projections φV : Pn 99K P2 ∪ ∪ C − 7 → C0 • C curve in Pn . • V a linear subspace in Pn of dimension (n − 3), called the centre of projection. • C 0 the image of C in P2 .
Torgunn Karoline Moe
Cuspidal curves
A special curve and some of its properties • The curve Cn in Pn given by
Introduction Background Cuspidal curves Cremona Projections
(s n : s n−1 t : . . . : st n−1 : t n ). is called the rational normal curve of degree n. • This is a smooth curve, and all points have the same
properties. • For each pair (s : t) corresponding to a point on Cn ,
Conjecture
the curve has a unique tangent line given by the linear span of
Real cusps
References
ns n−1 (n − 1)s n−2 t . . . t n−1 0 . 0 s n−1 . . . (n − 1)st n−2 nt n−1
• The union of all the tangents is a surface called the
tangent developable T of Cn .
Torgunn Karoline Moe
Cuspidal curves Introduction Background
• Two points on Cn define a unique line in Pn .
Cuspidal curves
• Such a line is called a secant line.
Cremona
• The union of all the secant lines is a threefold called
Projections Conjecture Real cusps References
the secant variety S of Cn . • The tangent developable is contained in the secant
variety.
Torgunn Karoline Moe
Cuspidal curves Introduction Background Cuspidal curves Cremona Projections Conjecture Real cusps References
Cuspidal projections • A rational cuspidal plane curve C 0 of degree n can be constructed from Cn using a suitable centre of projection V : • V can not intersect the curve Cn . • That would reduce the degree of C 0 . • V must intersect the tangent developable T . • Get one cusp on C 0 for every intersection point of V and T . • V can not intersect the secant variety S outside T . • That would give other singularities.
• We use Cn and the kernel of V written on matrix
form to find the parametrization of C 0 . • The defining polynomial of C 0 can be found by
eliminating s and t.
Torgunn Karoline Moe
Cuspidal curves Introduction Background Cuspidal curves Cremona Projections Conjecture Real cusps References
The cuspidal cubic • C3 = (s 3 : s 2 t : st 2 : t 3 ). • The centre of projection V is a point on the tangent
developable. • Can choose V = (0 : 1 : 0 : 0). • Parametrization – C 0 = (s 3 : st 2 : t 3 ). • Defining equation – C 0 = V(y 3 − xz 2 ).
Torgunn Karoline Moe
Cuspidal curves Introduction Background
The tricuspidal quartic • C4 = (s 4 : s 3 t : s 2 t 2 : st 3 : t 4 ). • The centre of projection V is a line which intersects
Cuspidal curves Cremona Projections Conjecture Real cusps References
the tangent developable in three points. • V can be found using the appropriate conditions.
V =
2 1 0 0 0 . 0 0 0 1 2
• C 0 = (s 3 t − 12 s 4 : s 2 t 2 : t 4 − 2st 3 ). • C 0 = V(8xy 3 − 3y 4 + 12xy 2 z − 4y 3 z + 4x 2 z 2 ).
Torgunn Karoline Moe
Cuspidal curves Introduction
Rational cuspidal curves of degree 5
Background
# Cusps
Cuspidal curves
1
Cremona Projections
2
Conjecture
3
Real cusps
4
References
Curve C1 C2 C3 C4 C5 C6 C7 C8
Cuspidal configuration (4) (26 ) (3, 2), (22 ) (3), (23 ) (24 ), (22 ) (3), (22 ), (2) (22 ), (22 ), (22 ) (23 ), (2), (2), (2)
# Curves 3 – ABC 1 2 – AB 1 1 1 1 1
Torgunn Karoline Moe
Cuspidal curves
Conjecture
Introduction
• There is only one rational cuspidal plane curve with
Background
more than three cusps – the curve of degree 5 with cuspidal configuration [(23 ), (2), (2), (2)]. • [Piontkowski (2007)] The only tricuspidal curves are
Cuspidal curves
• [Fenske, Flenner & Zaidenberg (1996-1999)]
Cremona
Series I II III
Projections Conjecture
m ¯p (d − 2) (d − 3, 2a ) (d − 4, 3a )
m ¯q (2a ) (3a ) (4a , 22 )
m ¯r (2d−2−a ) (2) (2)
For d d ≥4 d ≥5 d ≥7
• The curve of degree 5 with cuspidal configuration
[(22 ), (22 ), (22 )].
Real cusps References
d d 2a + 3 3a + 4
Result • [Tono (2005)] A rational cuspidal curve has ≤ 8
cusps.
Torgunn Karoline Moe
Cuspidal curves Introduction Background Cuspidal curves Cremona
Can all cusps be given real coordinates? • By the conjecture, a rational cuspidal curve will never have more than four cusps. • A linear transformation can move three cusps to real
coordinates. Projections Conjecture Real cusps References
• There is only one curve left to consider,
[(23 ), (2), (2), (2)].
Torgunn Karoline Moe
Cuspidal curves
• For real cuspidal curves, we have a modified Introduction Background Cuspidal curves
Klein–Schuh equality involving the curve C , its cusps and its dual curve C ∗ . d−
X real cusps on C
Cremona Projections Conjecture Real cusps References
(m − 1) = d ∗ −
X real cusps on
(m∗ − 1) C∗
• The dual of [(23 ), (2), (2), (2)] is the quartic [(23 )]. • Assuming the quintic is a real curve, and assuming
its cusps have real coordinates leads to a contradiction of Klein–Schuh. • Can the defining polynomial of [(23 ), (2), (2), (2)]
have complex coefficients while the cusps have real coordinates?
Torgunn Karoline Moe
Cuspidal curves Introduction
References T. Fenske Rational cuspidal plane curves of type (d, d − 4) with χ(ΘV hDi) ≤ 0.
Background Cuspidal curves Cremona Projections Conjecture
H. Flenner, M. Zaidenberg On a class of rational cuspidal plane curves. H. Flenner, M. Zaidenberg Rational cuspidal plane curves of type (d, d − 3). M. Namba. Geometry of projective algebraic curves.
Real cusps References
J. Piontkowski. On the Number of the Cusps of Rational Cuspidal Plane Curves. K. Tono On the number of the cusps of cuspidal plane curves.
